# Tagged Questions

130 views

### Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
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I'm currently in Real Analysis 2, and while I'm doing pretty well, it's just so much time and effort to finish all the problem sets and do well on the exams. I only have a few math classes left for ...
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### An ABC soft question about epsilon-delta argument

Someone told me that some textbooks present epsilon-delta argument somewhat misleadingly. For example, consider the simplest one: the convergence of the sequence $(1/n)_{1}^{\infty}$ to $0$. These ...
115 views

### Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
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### Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
125 views

### How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier analysis. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. ...
103 views

### Missing connection in real analysis?

I have long thought of real analysis as the consequences of mapping numbers to real numbers, thereby creating infinite sequences and series. From there, I can get to fairly elementary applications ...
2k views

### Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
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### Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
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### How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
531 views

### What level of rigour is expected in Real Analysis?

I fail to find a duplicate. I am wondering what level of rigour is needed in a typical undergraduate course in Real Analysis. To clarify my question, I provide an exercise from Rudin and my proposed ...
5k views

### What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
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### Does $f(x)\,dx$ denote multiplication of $f(x)$ by $dx$? [duplicate]

In the integral form $\int \! f(x) \, \mathrm{d}x$ does $f(x)\,\mathrm{d}x$ can be seen as a multiplication of $f(x)$ and $\mathrm{d}x$?
438 views

### I am failing at Analysis again. [closed]

I am stuck at this very hard course again. I don't want to put blames but if you were to ask me, what is going wrong I would say: The professor teaches too bluntly The book is too hard to ...
203 views

### How to think of zeros of the derivative of a holomorphic funcion?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, then, for example, if $f'(x_0)=0$ and $f'$ is again differentiable at $x_0$ and $f''(x_0)\neq 0$, then $f$ has a maximum or minimum at $x_0$. ...
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### Prove formally that $a \le x \le a \Rightarrow x = a$ for $x, a \in \mathbb R$ not using contradiction.

Prove formally that $a \le x \le a \Rightarrow x = a$ for $x, a \in \mathbb R$ not using contradiction. I've been thinking how to prove the above statement not using contradiction. My idea for a ...
649 views

### Why is analysis called “analysis”?

Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the ...
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### What is the best way to tell people what Analysis is about?

What is the best way to tell people what Analysis is about? I am currently taking Analysis course. However, I am really having a big difficulty explaining to people what Mathematical Analysis is ...
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### Prove that $\int_{-\pi}^{\pi}$ $\frac{d\theta}{1+\sin^2\theta}$ = $\pi\sqrt{2}$ using the method of Residues

Prove that $$\int_{-\pi}^{\pi}\frac{d\theta}{1+\sin^2\theta} = \pi\sqrt{2}$$ using the method of Residues How do I do this? I know I need it from $0$ to $2\pi$ but I don't know how to modify it!! ...